Comptes Rendus
Review article - Dynamical systems
The universal bound property
Alain Haraux1
1Laboratoire Jacques-Louis Lions, Sorbonne University, Pierre and Marie Curie Campus, 4 Pl. Jussieu, 75005 Paris, France
Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1499-1506

This survey paper is devoted to the question of universal bounds, independent of the initial state, for the trajectories of some nonlinear semi-groups and even more general processes. In the case of second order ODEs, rather surprisingly, it turns out that dissipation alone is not enough to produce such a property, and nonlinear elastic forces result in universal boundedness only when they dominate the damping in a very precise sense.

Cet article de synthèse est consacré à la question des bornes universelles, indépendantes de l’état initial, pour les trajectoires de certains semi-groupes non linéaires ou même de processus plus généraux. Dans le cas des équations différentielles ordinaires du second ordre, il s’avère, de manière assez surprenante, que la dissipation à elle seule ne suffit pas à garantir une telle propriété et que les forces élastiques non linéaires entraînent une bornitude universelle uniquement lorsqu’elles dominent l’amortissement d’une manière très précise.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.796
Classification: 34A40, 34D23, 34E10, 34G20, 35L71, 37L15
Keywords: Evolution equations, second order, damping term, restoring force, energy bounds
Mots-clés : Équations d’évolution, second ordre, terme d’amortissement, force de rappel, majorations d’énergie

Alain Haraux  1

1 Laboratoire Jacques-Louis Lions, Sorbonne University, Pierre and Marie Curie Campus, 4 Pl. Jussieu, 75005 Paris, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Alain Haraux. The universal bound property. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1499-1506. doi: 10.5802/crmath.796

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